Thermonuclear inverse magnetic pumping power cycle for stellarator reactor

ABSTRACT

The plasma column in a stellarator is compressed and expanded alternatively in minor radius. First a plasma in thermal balance is compressed adiabatically. The volume of the compressed plasma is maintained until the plasma reaches a new thermal equilibrium. The plasma is then expanded to its original volume. As a result of the way a stellarator works, the plasma pressure during compression is less than the corresponding pressure during expansion. Therefore, negative work is done on the plasma over a complete cycle. This work manifests itself as a back-voltage in the toroidal field coils. Direct electrical energy is obtained from this voltage. Alternatively, after the compression step, the plasma can be expanded at constant pressure. 
     The cycle can be made self-sustaining by operating a system of two stellarator reactors in tandem. Part of the energy derived from the expansion phase of a first stellarator reactor is used to compress the plasma in a second stellarator reactor.

CONTRACTUAL ORIGIN OF THE INVENTION

The U.S. Government has rights in this invention pursuant to ContractNo. DE-AC02-76CH03073 between the U.S. Department of Energy andPrinceton University.

BACKGROUND OF THE INVENTION

The present invention relates generally to a method for generatingelectricity from a fusion reactor and more particularly to a method fordirect conversion of alpha-particle energy into electricity in astellarator reactor.

The guest to tap the energy of nuclear fusion by employing magneticfields to confine an ultrahot plasma and generating electric power hasbeen in progress for more than three decades.

Several toroidal magnetic confinement fusion devices have been proposed.One such device is the tokamak where a toroidal current induced insidethe plasma both heats the plasma and provides the poloidal magneticfield. There are several drawbacks, however, associated with thetokamak. The large plasma current needed for confinement in a tokamak,carries a large free energy that can be tapped by instabilities whichdestroy the confinement. Another problem associated with the plasmacurrent is that it must be maintained by some means other than atransformer, since otherwise the pulse length is limited by the numberof volt-seconds in the transformer windings.

Another toroidal confinement machine is the stellarator, where thepoloidal field is produced externally to the plasma by current-carryingconductors wound helically around the torus. This configuration does notrequire the large plasma current needed in a tokamak. Stellarators arecapable of achieving betas several times greater than betas achievablein a tokamak. Stellarators are also capable of steady-state operation.

A major problem in the design of a commercial fusion stellaratorreactor, as with other conventional power sources, is the conversion ofthe thermal energy produced into electrical energy.

Conventional designs call for the use of the thermal energy produced byfusion reactions to convert water to steam. The steam is used in adynamic conversion processes to drive turbines and turbogenerators. Thisdynamic conversion process requires turbines, pumps, generators, largecooling systems and extensive piping systems. This auxiliary equipmentis expensive, unreliable and relatively inefficient.

Direct energy conversion techniques for tokamaks have been suggested inFusion Energy Conversion, by George H. Milley, published by the AmericanNuclear Society, 1976. The problems associated with tokamaks, however,have been discussed above.

Therefore, in view of the above, it is an object of the presentinvention to provide a novel cycle of operation for a stellarator fusionreactor.

It is another object of the present invention to provide a novel cycleof operation for a stellarator fusion reactor for directly convertingfusion energy into electrical energy.

It is another object of the present invention to provide a novel cycleof operation of a stellarator fusion reactor which may be used inadvanced neutron-beam fueled reactors.

Is is a further object of the present invention to provide a cycle ofoperating two stellarators in tandem, such that the cycle isself-sustaining.

It is still another object of the present invention to provide astellarator reactor capable of directly generating electricity.

It is still a further object of the present invention to provide astellarator reactor system which is self-sustaining.

Additional objects, advantages and novel features of the invention willbecome apparent to those skilled in the art upon examination of thefollowing or may be learned by practice of the invention. The objectsand advantages of the invention maybe realized and attained by means ofthe instrumentalities and combinations particularly pointed out in theappended claims.

SUMMARY OF THE INVENTION

To achieve the foregoing and other objects in accordance with thepurpose of the present invention, as embodied and broadly describedherein, the method of this invention may comprise a three step processin which the minor radius of a stellarator is compressed and expanded.In the first step an ignited plasma is in thermal balance. The plasma iscompressed adiabatically and the plasma β decreases. In the next stepthe plasma volume is kept constant and the plasma temperature and 62 aredriven up by the excess thermonuclear alpha-particle heating. As βapproaches the maximum β attainable, the rate of energy loss increasesuntil it balances the alpha-particle heating power and the plasma isagain in a state of thermal balance. In the final step the plasma isexpanded back to its original radius. When the plasma expands thecorresponding pressure is higher than the corresponding pressure duringcompression since β stays at β_(c) during the expansion. Therefore,negative work is done on the plasma over the complete cycle. This workmanifests itself as a back-voltage in the toroidal field coils anddirect electrical energy is obtained from this voltage.

As an alternate cycle, net work can also be done on the external systemby allowing the plasma to expand at a constant pressure in the secondstep of the method, rather than keeping the plasma at constant volume.

A magnetic confinement fusion reactor for directly generatingelectricity using the methods of the present invention comprises: avacuum vessel; helical stabilizing coils; toroidal confining coils;means for generating a current through the toroidal coils; means forcompressing a plasma disposed in the vacuum vessel; means formaintaining the volume of the plasma constant; means for expanding theplasma and means for transmitting current generated by the plasma fromthe toroidal coils.

By operating two or more reactors in tandem, such that part of theenergy produced by one reactor is used to compress and/or maintain aconstant plasma volume in a second reactor the cycle can be madeself-sustaining.

The present invention provides a method and apparatus for obtainingelectrical energy directly from a stellerator fusion reactor. Therefore,the present invention obviates the need for any of the interveningmachinery associated with a turbogenerator.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic representation of a cross-section of a stellaratorreactor at various stages during the thermonuclear inverse magneticpumping power cycle.

FIG. 2 is a graph showing a comparison of heating power with powerlosses for the plasma in the stellarator with machine parameters asgiven in Table I.

FIG. 3 shows the net work done normalized to machine dependentparameters.

FIG. 4 is a pressure-volume (P-V) schematic for the cycles of thepresent invention.

FIG. 5 shows a graph of the thermal efficiency versus the compressionratio for the cycles of the present invention.

FIG. 6 is a temperature-entropy (T-S) schematic for the cycles of thepresent invention.

FIG. 7 shows a temporal evolution of the plasma temperature afteradiabatic compression, for the reactor with parameters given in Table I.

FIG. 8 is a schematic representation of currents in coils and electricalenergy flow between reactors as a function of time for a coupled reactorsystem.

FIG. 9 is a schematic representation of two stellarator reactors coupledtogether such that the operation of the system is made self-sustaining.

DETAILED DESCRIPTION OF THE INVENTION

Reference will now be made to the present preferred embodiments of thepresent invention, an example of which is illustrated in theaccompanying drawings. FIG. 1 illustrates the power cycle of the presentinvention, for direct conversion of alpha-particle energy to electricityfor a stellarator reactor. This power cycle provides an alternativescheme in energy conversion for deuterium-tritium (D-T) fueled reactorsand could become an important method in energy conversion for advancedneutron-lean fueled (e.g. D-He³) reactors. The direct energy conversionis achieved by alternately compressing and expanding the plasma minorradius of a stellarator reactor 10. The cycle may be used in astellarator reactor as shown by reactor 10 in FIG. 9. the cycle will nowbe discussed with reference to a single stellarator reactor 10. (Adescription of a two-reactor system is described subsequently.) Reactor10 includes external toroidal magnetic confining coils 12, helicalstabilizing coils 13, and vacuum vessel 19. A generator 15 provideselectrical energy to coils 12 via line 34. Control means 37 operativelyassociated with lines 32 and 34 engage and disengage load 17 andgenerator 15 from toroidal coils 12 in response to plasma parameters.The cycle is composed of three stages.

In a preferred embodiment of the present invention, ignited plasma 16 isthermally stable and in thermal balance at stage 0. (Conditions for anignited plasma may be found in the published literature. For example,reference is made to Controlled Thermonuclear Reactions by SamuelGlasstone and Ralph Lovberg published by D. Van Nostrand Co., Inc.,Princeton, N.J., 1960.) The plasma β is slightly above β_(c).[β=(P/B_(i) ² /8π), i.e. the ratio of plasma kinetic to the internalmagnetic field pressure, and β_(c), the maximum attainable β, is set bythe onset condition for MHD ballooning instabilities.] During thecompression phase (from stage 0 to stage 1), plasma 16 is compressedadiabatically by increasing the external toroidal compressed magneticfield strength 14 generated by toroidal coils 12 and the plasma βdecreases. At stage 1, the end of the compression phase, β is less thatβ_(c) and the plasma 16 is no longer in thermal balance since thethermonuclear alpha-particle heating power exceeds the rate of energyloss (due to neoclassical transport, turbulent convection, andBremsstrahlung radiation). During the heating phase (from stage 1 tostage 2), as the plasma temperature is driven up by the excess heatingpower, the plasma volume is kept constant by further increasing theexternal field strength 14. As β approaches β_(c), the rate of energyloss increases until it balances the alpha-particle heating power when βreaches β_(c). The plasma 16 is again in thermal balance and isthermally stable. This is stage 2 of the cycle.

To complete the cycle, the external field 14 is then reduced so that theplasma 16 expands back to its original radius during the expansion phase(from stage 2 to stage 0). When the plasma 16 expands, the plasma βtends to increase. However, β is already at β_(c) at stage 2, henceballooning instabilities force β to stay at β_(c) through turbulentconvection during the entire expansion phase. As a result, the plasmapressure during the expansion is higher than the corresponding pressureduring the compression. Therefore, negative work is done on the plasma16 during a complete. cycle. Note that if there is no β limit, then morework can be obtained during a complete cycle. This work manifests itselfas a back-voltage in the toroidal field coils 12 and direct electricalenergy is obtained from this voltage. By operating two or more reactorsin tandem, the cycle can be made self-sustaining.

As an alternative cycle, net work is done on the external system byletting the plasma expand at constant pressure between stages 1 and 2,rather than keeping the plasma at constant volume. These two methods(heating phase at constant plasma volume and at constant plasmapressure) are discussed below as the preferred embodiments of thepresent invention. As will be recognized by those skilled in the art,there are many other possible methods, that can also result in net workdone, e.g. the plasma minor radius varies sinusoidally during the cycle.

If the external magnetic field 14 is considered to be a "piston," thenthe external system absorbs work done by the pumping action of theplasma 16 on the piston. This action will be referred to as "magneticpumping". It is "inverse magnetic pumping" over a full cycle becausework is done on the external system by the plasma 16. Note that thisconcept of magnetic pumping differs from the traditional definitionwhich refers to "gyro-relaxation,".

The detailed physical processes at, and between, various stages of thecycle are analyzed below. To simplify the analysis, the plasma column ismodeled by a straight cylinder. The plasma density, temperature, and themagnetic field 18 are assumed to be uniform across the minor radiusexcept for a sharp discontinuity across the boundary between the plasma16 and the confining vacuum magnetic field 14. Throughout this analysis,the plasma is assumed to behave like an ideal MHD fluid.

Due to the nonlinear temperature dependence of the alpha-particleheating power term and the power loss terms, the thermonuclear plasma ina stellarator can have more than one thermal equilibrium, i.e., thethermonuclear power can balance the power losses at differenttemperatures, or have no equilibrium at all. A qualitative discussionabout the possible thermal equilibria and their stability is given here.The reactor parameters listed in Table I are used in the discussionhere.

                  TABLE I.                                                        ______________________________________                                        STELLARATOR REACTOR PARAMETERS                                                ______________________________________                                        Plasma density (cm.sup.-3)                                                                             6 × 10.sup.14                                  Plasma temperature (keV)                                                                              10                                                    Toroidal magnetic field (KG)                                                                          50                                                    Plasma β (%)       19.3                                                  Major Radius (m)        10                                                    Minor Radius (m)         2                                                    εh               0.1                                                  ______________________________________                                    

For a fixed plasma density (n=6×10¹⁴ cm⁻³), the thermonuclearalpha-particle heating power per unit volume is plotted versus plasmatemperature as the solid curve in FIG. 2. In the same figure, the powerloss per unit volume [due to neoclassical diffusion (discussed below)and Bremstrahlung radiation] as a function of temperature appears as thedashed curve. (The power loss due to turbulent convection is notincluded in this dashed curve but its effect on thermal equilibrium isdisclosed below). FIG. 2 indicates that there are two equilibriumsolutions. The first (point A) is thermally unstable. If the plasmatemperature decreases slightly, the neoclassical diffusion andBremsstrahlung radiation losses will begin to dominate and hence theplasma temperature will decrease to zero. On the other hand, if there isa slight increases in plasma temperature, the alpha-particle heatingpower will dominate and thus the plasma temperature will proceed on athermal excursion until the loss rate matches the heating power at astable thermal equilibrium, which is the second equilibrium solution(point C). (Similar discussions for tokamak reactors have been given byConn, Fusion, Academic Press, N.Y. (1981), and by Kolesnichenko andReznik, Plasma Physics and Controlled Nuclear Fusion Research, 1976,Proc., 6th Int. Conf. Berchtesgaden, 1976, vol. 3, IAEA, Vienna 1977,347.) Thus, to start a stellarator reactor, it is only necessary toraise the plasma temperature to the unstable equilibrium at point A,even if the desired operating condition at the stable equilibrium is ata higher temperature. It is interesting to note that if a₀ is reduced to1.5 m, but all the other reactor parameters remain unchanged, then thestable thermal equilibrium occurs at some temperature between 7-8 keV.The stable thermal equilibrium (point C) in FIG. 2 requires a plasmatemperature of approximately 11 keV. If β_(c) =19.3%, then β at point Cexceed β_(c). As a result, pressure driven turbulent convection willincrease the power loss, raise the loss curve, and force the stableequilibrium to occur at a lower plasma temperature (point B) such thatβ≅β_(c). (As β approaches β_(c), the plasma will follow the broken curvein FIG. 2).

After adiabatic compression, due to the increase in both thealpha-particle heating power and the power losses, the solid curve inFIG. 2 shifts upward and the equilibrium point shifts toward higherplasma temperature and power (the upper-right hand corner). As discussedbelow, the increase in the alpha-particle heating power is greater thanthe increase in the power losses. This implies that, immediately aftercompression the plasma temperature is located somewhere between thecorresponding thermal equilibria (points A and B) of the shifted curves.Consequently, the plasma temperature continues to rise until it reachesa new stable equilibrium.

Before the compression, the ignited plasma 16 is in thermal balance, ata stable thermal equilibrium. The plasma β is slightly above β_(c).Thus, the plasma pressure gradient exceeds the critical pressuregradient by a small amount. Turbulent convective cells are driven bythis excess pressure gradient and carry part of the outward particle andenergy fluxes as discussed by Ho and Kulsrud, PPPL-2251; the rest of thefluxes are carried by neoclassical transport as discussed by Ho andKulsrud, PPPL-2253. This is stage 0 in the power cycle. The plasma andmagnetic field parameters are:

    ______________________________________                                        Plasma β          β.sub.0  = β.sub.c,                          Plasma pressure        P.sub.0,                                               Plasma minor radius:   a.sub.0,                                               Internal magnetic field:                                                                             B.sub.i0,                                              External magnetic field:                                                                             B.sub.e0.                                              ______________________________________                                    

The subscript 0 refers to the physical quantities at stage 0.

During the compression phase from stage 0 to stage 1, the plasma 16 iscompressed adiabatically as the external toroidal magnetic fieldstrength 14 is increased. Adiabatic compression occurs if thecompression time is shorter than the burn time (the time for thethermonuclear alpha-particles to increase the plasma temperature by anamount comparable to itself). It is important for the compression to beadiabatic because otherwise, the alpha-particles could heat up theplasma and more work would be required during the compression phase. Onthe other hand, as explained below, the compression phase must becarried out slowly compared with τ_(i) and τα (the 90° deflection timesfor ions and alpha-particles) so that the compression isthree-dimensional.

The plasma β decreases during the compression because the internalmagnetic field pressure increases faster than the plasma kineticpressure. This can be shown explicitly if the plasma β is expressed interms of P₀ and B_(i0). Using the adiabatic compression law (PV.sup.γ=const with γ=5/3), the definition of β, and the frozen flux condition,the plasma β can be expressed as ##EQU1## Here r_(v) (t)=a₀ /a(t) is thecompression ratio. During the compression, r_(v) increases from unity toa₀ /a, and thus β decreases. If β_(c) is roughly constant, then theplasma β is below β_(c) after compression. This is stage 1 in the powercycle. The plasma and magnetic field parameters are:

    ______________________________________                                        Plasma β:       β.sub.1  < β.sub.0  = β.sub.c,            Plasma pressure:     P.sub.1  > P.sub.0,                                      Plasma minor radius: a.sub.1  < a.sub.0,                                      Internal magnetic field:                                                                           B.sub.i1  > B.sub.i0,                                    External magnetic field:                                                                           B.sub.e1  > B.sub.e0.                                    ______________________________________                                         The subscript 1 refers to the physical quantities at stage 1.

After the adiabatic compression, the increase in alpha-particle heatingpower per unit volume is larger than the increase in neoclassical energydiffusion and Bremsstrahlung radiation losses per unit volume. Also,turbulent convective transport can be assumed to be absent aftercompression since β is less than β_(c). This imbalance between theheating power and the power loss causes the plasma temperature toincrease while the plasma volume is kept constant by further increasingthe external magnetic field strength 14. The increase in temperature canbe understood in more detail from the volume-averages energy balanceequation ##EQU2## The first term on the right-hand side of this equationis the heating power from the thermonuclear alpha-particles. The firstterm inside the bracket is the energy loss rate due to neoclassicaldiffusion, i.e. ##EQU3## As discussed by Ho et al., PPPL-2253, theelectron energy confinement time is ##EQU4## and the ion energyconfinement time is

    τ.sub.E.sup.i ≅τ.sub.E.sup.e.

Here the minor radius a_(m) and the major radius R_(m) are in meters, B₄is in 10KG, electron or ion density n₁₄ is in 10¹⁴ cm⁻³, plasmatemperature T₄ is in 10 keV, ε_(h) is the depth of the helical magneticwell caused by external helical windings, and the normalized ambipolarelectric field strength C≅(e/T)(∂Φ/∂r)n(∂n/.differential.r)⁻¹ can beapproximated by unity. The second term inside the bracket in Eq. (2) isthe convective energy loss rate. The last terms is the energy loss ratedue to Bremsstrahlung radiation and is equal to cn² √T, with "c" aconstant. Note that density is not a function of time during the heatingphase since the plasma volume is held fixed. Also note that before thecompression, thermal balance means that dT/dt=0. The ratio of the valueof each term on the right-hand side of Eq. (2) after the adiabaticcompression to the value of the corresponding term before thecompression will now be studied. Because of the increase in plasmadensity and temperature after the adiabatic compression, the rate ofthermonuclear alpha-particle energy production is larger than thatbefore the compression by the ratio ##EQU5## Here 3.5 MeV is the energyof an alpha-particle generated from D-T reactions, and <σv<₀ and <σv>₁are the Maxwellian reactivities at stage 0 and stage 1, respectively. Ifr_(v) =1/0.6 amd T₀ =10 keV, then T₁ =19.8 keV. Using the formula for<σv>_(D-T) given in NRL Plasma Formulary (D.L. Book Ed.), Naval ResearchLaboratory, Washington, D.C. (1980), Eq. (5) has a value of 32.1. Toobtain the ratio of the neoclassical energy loss rate per unit volumeafter the compression to that before the compression, we let P_(neo)≅3nT/.sub.τ^(e). Then, it can be shown that ##EQU6## which has a valueof 7.72 for r_(v) =1/0.6. The turbulent convective transport is assumedto vanish at the end of the compression phase since β drops below β_(c).Finally, the ratio of the Bremsstrahlung radiation energy loss rateafter compression to that before the compression is ##EQU7## which has avalue of 10.85 for r_(v) =1/0.6. From Eqs. (5)-(7), we can conclude thatalpha-particle heating will further increase the plasma temperatureafter the adiabatic compression since the relative increase in thealpha-particle heating power is larger than the relative increase in theenergy loss rates. This phenomenon of plasma temperature rise aftercompression is of fundamental importance to the success of the powercycle of the present invention, since the plasma must have a higherpressure during the expansion phase than during the compression. As thetemperature rises, the plasma β increases since the plasma volume isheld fixed. As β approaches β_(c), the convective energy loss re-emergesand the convective energy loss rate gradually catches up with theincrease in the alpha-particle heating power. After the compression, theplasma temperature asymptotically approaches the limit at which β=β_(c)in a characteristic time defined as the "thermal relaxation time" (seeFIG. 7). At stage 2 in the power cycle, the plasma is again thermallystable and in thermal balance, at a stable equilibrium. The plasma andmagnetic field parameters are:

    ______________________________________                                        Plasma β          β.sub.2  = β.sub.c,                          Plasma pressure:       P.sub.2  > P.sub.1,                                    Plasma minor radius:   a.sub.2  = a.sub.1,                                    Internal magnetic field:                                                                             B.sub.i2  = B.sub.i1,                                  External magnetic field:                                                                             Be.sub.2  > B.sub.e1.                                  ______________________________________                                    

To complete the cycle, the external magnetic field 14 is decreased sothat the plasma 16 expands back to its original minor radius. This isthe expansion phase. When the plasma expands, the plasma β tends toincrease. [According to Eq. (1), β increases as r_(v) decreases].However, β is already at β_(c) at stage 2, hence ballooninginstabilities force β to stay at β_(c) (recall that β_(c) is assumed tobe a constant) through turbulent convection during the entire expansionphase. Consequently, the plasma pressure during the expansion phase canbe expressed as ##EQU8## Upon applying the frozen flux condition, thisexpression can be written as ##EQU9## which varies exactly as though ithas an adiabatic index γ=2. As a result, the plasma pressure during theexpansion is higher than the corresponding pressure during thecompression. Therefore, negative work is done on the plasma 16 during acompleter cycle. This work manifests itself as a mean back-voltage inthe toroidal field coils 12, and direct electrical energy is obtainedfrom this voltage. This electrical energy may be transfered to a load 17via line 32. Using the pressure-volume (P-V) relation, the amount ofwork done on coils and the thermal efficiency of the cycle arecalculated below. It is now clear that if the compression phase has beencarried out faster than τ_(i), then the compression would betwo-dimensional (γ=2) and the pressure variation during the compressionwould follow Eq. (8) instead of PV^(5/3) =const. Consequently, no network would be done on the external system since the plasma pressureduring the compression phase would equal the corresponding pressureduring expansion.

Finally, the cycle described here satisfies the Kelvin-Planck statementof the second law of thermodynamics by losing heat (obtained fromthermonuclear alpha-particles) to the outside through turbulentconvection.

During the heating phase, between stages 1 and 2, the plasma can eitherbe held at constant volume or be allowed to expand at constant pressurewhile receiving energy from the thermonuclear alpha-particles. In acomplete cycle, both methods result in net work done on the externalsystem. The work done and thermal efficiency for each of these methodsare now calculated.

The preferred embodiment of the thermonuclear inverse magnetic pumpingpower cycle with heating phase at constant plasma volume is discussedfirst. To calculate the amount of work delivered during this cycle, itis only necessary to consider the work performed by the plasma duringthe compression and expansion phases. No work is performed during theheating phase because the plasma volume is constant.

The work done on the plasma during the adiabatic compression is##EQU10## Invoking P₀ =β_(c) B_(i0) ² /8π and the pressure balanceequation P₀ +B_(i0) ² 8π=B_(e0) ² /8π, P₀ can expressed as ##EQU11##Using this expression, ₀ W₁ can be written as ##EQU12## where L=2πR isthe length of the system. The work done by the plasma during theexpansion phase is calculated by using Eqs. (8) and (9). The result is##EQU13## Therefore, the amount of net work done on the coils during acomplete cycle is ##EQU14## This expression shows that either a highercompression ratio or a larger β_(c) will give a larger amount ofdelivered work. The net work done normalized by the machine dependentparameters (1/8)a₀ ² Lβ_(c) [1/(1+β_(c))]B_(e0) ² is ##EQU15## which isplotted versus the compression ratio in FIG. 3. Although it is desirableto operate the cycle at a high compression ratio, the attainablecompression ratio may depend on the highest achievable toroidal magneticfield. This field is limited by the allowable static and dynamic loadingon the machine structure. The amount of work delivered by the cycle canbe represented by the area of the triangle 0-1-2 in the pressure-volumediagram shown if FIG. 4.

The performance of this cycle is characterized by the thermal efficiencyη_(th) which is defined as ##EQU16## where O_(H) represents the heatobtained by the cycle (engine) from a heat source (thermonuclearalpha-particles). To calculate the heat received by the plasma duringthe heating phase, use the first law of thermodynamics,

    Q.sub.H =.sub.1 U.sub.2 +.sub.1 W.sub.2.                   (14)

Here ₁ U₂ is the change in plasma internal energy between stages 1 and 2and can be expressed as

    .sub.1 U.sub.2 =3 a.sub.0.sup.2 Ln.sub.0 (T.sub.2 -T.sub.1).

Since the plasma volume is held fixed during the heating phase, Q_(H) =₁U₂. From P₂ =β_(e) B_(i2) ² /8π, the frozen flux condition, and theadiabatic compression law, it can be shown that ##EQU17## Using Eqs.(12) and (15), the thermal efficiency can be expressed as ##EQU18## Theimportant thing to note is that the efficiency of this cycle is afunction only of the compression ratio and increases with it. Thethermal efficiency is plotted versus the compression ratio in FIG. 5. Asthe compression ratio approaches infinity, the thermal efficiencyapproaches a value of 2/3.

The thermal efficiency can be visualized by looking at the temperatureentropy (T-S) diagram for the cycle, FIG. 6. The thermal efficiency isthe ratio of the area of the triangle 0-1-2 to the area beneath the path1-2.

Up to this point, the discussions have been restricted to the case inwhich there is a limiting β. However, the cycle will become moreefficient and more work can be obtained if β_(c) is very high or ifthere is actually no β limit (this situation may occur in a heliac). Forthis case the work required to compress the plasma is given by Eq. (10).The plasma pressure at stage 2 may now reach a higher value than in thecase where there is a β limit. [Note that turbulent convection is absenthere since there is no β limit.] During the expansion, the plasmapressure variation follows PV^(5/3) =const since β will no longer beprevented from rising. Hence, ##EQU19## where the subscript 0' denotesconditions at stage 0'--the moment when the expansion phase is completedand the plasma volume returns to its original value at stage 0. At stage0', P_(0') >P₀ and the power losses are greater than the alpha-particleheating power. Thus, the plasma temperature will decrease. While thetemperature is decreasing, the external magnetic field is reduced inorder to keep the plasma at constant volume and heat diffuses to theoutside by neoclassical transport. The cycle is completed when theplasma temperature and external magnetic field return to their originalvalues at stage 0 (point C in FIG. 2).

The net work is ##EQU20## where T_(0') can be related to T₂ using theadiabatic law.

The thermal efficiency is ##EQU21## which is the Otto cycle efficiency.Note that this cycle is identical to the Otto cycle.

The preferred embodiment of the thermonuclear inverse magnetic pumpingpower cycle with heating phase at constant plasma pressure is nowdiscussed. In this power cycle, the compressed plasma is allowed toexpand at constant pressure until the plasma β reaches β_(c). At thispoint (stage 2), the plasma radius is between the minor radius beforethe compression and that after the compression. Next, the plasma columnis expanded further with β staying at β_(c) until the minor radiusreaches the pre-compression value. The cycle is now complete.

To calculate the amount of work delivered during this cycle, note thatthe work required to compress the plasma is the same as that of theprevious cycle, see Eq. (10). To calculate the work done by the plasmaon the confining field (₁ W₂ +₂ W₀), it is necessary to know the plasmaminor radius at the end of the constant pressure heating phase. Startingwith P₂ =β_(c) B² _(i2) /8π, using the frozen flux condition, and notingthat P₂ =P₁, it can be shown that

    a.sub.2 =a.sub.0 r.sub.v.sup.-5/6.                         (19)

The work done by the plasma on the coils during the constant pressureheating phase is ##EQU22## Using Eq. (9), the work done by the plasmaduring the final constant β_(c) expansion phase is ##EQU23## Therefore,the amount of net work done on the coils 12 during a completer cycle is##EQU24## where W_(norm) =2 r_(v) ⁴ 5^(/3) -(5/2)r_(v) ^(4/3) +1/2 isplotted versus the compression ratio in FIG. 3. Again, the amount ofwork delivered by this cycle is represented by the area of the triangle0-1-3 in the pressure-volume diagram, FIG. 4.

The thermal efficiency, which again depends only on the compressionration, is ##EQU25## The thermal efficiency is plotted versus thecompression ratio in FIG. 5. As the compression ratio approachesinfinity, the thermal efficiency approaches a value of 4/5. Now, thethermal efficiency is the ratio of the area of triangle 0-1-3 to thearea beneath the path 1-3 in FIG. 6. Thus, for the same compressionration, this cycle delivers less work but at a higher thermal efficiencythan the previous cycle. From the standpoint of reactor economics, it isprobably the net work done, rather than the thermal efficiency, that isimportant.

The power delivered by the power cycle via direct energy conversiondepends on the period of the cycle. One of the major factors thatgoverns the period is the thermal relaxation time, i.e. time for theplasma β to reach β_(c) after adiabatic compression. The rate of energytransfer of an individual alpha-particle to the background plasma 16 isnow considered. The time evolution of the plasma temperature after thecompression is also studied using the volume-averaged energy equationfor electrons. From the numerical solution of this equation, the thermalrelaxation time is estimated.

The rate of thermonuclear alpha-particle energy transferred to thebackground plasma is given by Trubnikov, Review of the Plasma Physics(M. A. Leontovich, Ed.), Consultant Bureau, New York (1965), vol. 1, inthe form ##EQU26## where ε.sub.α is the alpha-particle energy, andτ.sup.α/D and τ.sup.α/e are, respectively, the characteristic energytransfer time between the alpha-particle and electrons, and thecharacteristics energy transfer time between the alpha-particle andelectrons. In this analysis, we assume T =T_(e) =T_(i). To obtain thecharacteristic energy transfer time, not that after the compression (atstage 1), the plasma with stage 0 parameters give in Table I has n₁=16.7×10¹⁴ cm⁻³ and T_(i) =19.8 keV if r_(v) =1/0.6. Thus ##EQU27##where ε.sub.α (0)=3.5 MeV is the energy of an alpha-particle generatedfrom D-T reactions, and m_(e) and m.sub.α are the electron andalpha-particle mass, respectively. Using these limits, the expressionsfor τ.sup.α/e can be simplified accordingly. Trubnikov, cited above,showed that ##EQU28## For ε.sub.α (0)=3.5 MeV, it can be shown thatτ.sup.α/e =τ.sup.α/D /57. Thus, at the time of birth of analpha-particle, the rate of energy transfer to the electrons is aboutsixty times faster from an alpha-particle that to the deuterons. At thecritical energy given by ##EQU29## the rate of energy transfer from analpha-particle to the electrons is equal to that to the ions. At T₁=19.8 keV, ε.sub.αcrit =0.29 MeV. Thus, the thermonuclearalpha-particles lose most of their energy to electrons and for practicalpurposes, the alpha-particles energy transfer rate can be approximatedby ##EQU30## Using Eqs. (23) and (24), the characteristics energytransfer time can be expressed as ##EQU31## Substituting the values ofdensity and temperature of the plasma immediately after the compressionthat Eq. (26) gives τ.sup.α/e =0.05 sec. Note that using Eq. (4), it canbe shown that after the compression, the plasma energy confinement timeτ.sub.ε ≅0.32 sec. Hence τ.sup.α/e <τ.sub.ε.

To obtain the temporal evolution of the plasma temperature after thecompression, assume T_(e) =T_(i) and use the volume-averaged energybalance equation for the plasma: ##EQU32## In this equation, theturbulent convective energy loss is not included since we have assumedthat there is no convective loss for β<β_(c) (in reality, however,convective loss emerges gradually as β approaches β_(c) as was mentionedabove). Let the time at the end of the compression phase be zero. Then,the power deposited by the alpha-particles to the background plasma perunit volume, P (t), at any time t after the compression can be expressedas ##EQU33## Here the first term on the right-hand side of this equationrepresents the alpha-particle power deposition at any time t>0 by allthe alpha-particles generated in some intermediate time between 0≦t'≦t(0≧t'≧-∞); n₀ is the density before the compression, and dε.sub.α(t,t')/dt [given by Eq. (23)] represents the rate of energy transfer attime t, of an individual alpha-particle generated at some previous timet'. The factor (1+2r_(v) ²)/3 in the second term is the increase inalpha-particle energy due to two-dimensional adiabatic compression(compression is two-dimensional if the compression phase is carried outfaster than τ.sub.αi - and 90° deflection time between an alpha-particleand background ions).

Equation (27) is an integrodifferential equation, but it can beconverted into a second order differential equation by taking a timederivative. Using Eqs. (25) and (26), it can be shown that ##EQU34##Using this expression, we find that the time derivative of Eq. (28) canbe written as ##EQU35## where <σv>_(t) is the Maxwellian reactivity attime t. Then, after taking the time derivative of Eq. (27), using theabove expression for dPα/dt, and performing some straight forwardalgebra, it can be shown that This equation can be solved numerically.Two initial conditions are needed. The first one is

    T(0)=r.sub.v.sup.4/3 T.sub.0                               (30a)

from the adiabatic law. [Note that T(0) denotes the plasma temperatureat time 0 (stage 1) and T₀ denotes the plasma temperature at stage 0.]To determine the second initial condition dT/dt|_(t=0), we need to knowPα(0). If the adiabatic compression time is faster that τ.sub.αi, thenthe alpha-particle heating power per unit volume immediately after thecompression is increased from the corresponding heating power before thecompression by the ratio ##EQU36## Here 0⁻ denotes the time at thebeginning of the compression. The second initial condition for Eq. (29)can now be written as ##EQU37## Using the initial conditions given byEqs. (30a) and (30b), the solution of Eq. (29) for the reactor withparameters given in Table I with r_(v) =1/0.6 is obtained and plottedversus time as the solid curve in FIG. 7. This figure shows that theplasma temperature asymptotically approaches a limit of 35.5 keV.

If we let the thermal relaxation time for the plasma β to reach β_(c) bethe time that it takes the plasma temperature to reach 90% of thelimiting temperature at 35.5 kev, then this time is approximately 0.16,as illustrated in FIG. 7. This condition would represent the end of theheating phase.

Note that the thermal relaxation time is somewhat larger than τ.sup.α/e.In the limit that the relaxation time is much greater than τ.sup.α/e,Eq. (28) can be simplified by assuming <σv>_(t') ≈<σv>_(t). Furthermore,the second term on the right-hand side of Eq. (28) is generally smallcompared to the first term and thus can be neglected. The powerdelivered to the plasma now becomes ##EQU38## which is the instantaneousfusion power. This result simplifies Eq. (27) to a first-order equation.The temporal evolution of temperature described by this simplifiedequation is plotted versus time as the dashed curve in FIG. 7.

Finally, note that if the turbulent convective loss were included in thecalculation, then the resulting power loss would be higher due to theturbulent convection as β approaches β_(c) and the rate of temperaturerise would become slower. However, the thermal relaxation time wouldprobably be about the same, since turbulent convection also forces theasymptotic limit of the plasma temperatures to drop to a lower value atwhich β=β_(c).

To make the power cycle self-sustaining, part of the work done by theplasma on the coils during the expansion phase must be stored in orderto supply power for the next compression phase. However, the amount ofenergy to be stored is large compared with the net work done by theplasma during the cycle. Thus, even if the ratio of the energy lost(during the transfer to, plus the loss in the storage system) to thetotal energy transferred from the reactor to the storage system (thisratio is defined as the re-circulation inefficiency) is only a fewpercent, the total energy lost in recirculation may still exceed thework done during the cycle. Consequently, the power cycle may require alower re-circulation inefficiency than can be provided by conventionalenergy storage systems, e.g. capactive, inductive, and inertial(motor-generator-flywheel). To obviate this, in a preferred embodimentof the present invention two (or more) stellerator reactors are operatedin tandem. The major electrical energy loss in this system is theresistive dissipation in the coils 12, but this loss may be smallcompared with the net work performed by the cycle as is discussed below.

The reactor system of this preferred consists of two identicalstellerator reactors, 10 and 20 as shown in FIG. 9. Reactor 20 includesexternal magnetic confining coils 22, helical stabilizing coils 23 andand vacuum vessel 29. The cycle with heating phase at constant volumewill be used for discussion here, however, as will be recognized bythose skilled in the art other power cycles may also be used. Theoperation scenario is illustrated in FIG. 8.

When the plasma in reactor 10 is expanding, some of the resulting workis transferred via line 44 and used to compress the plasma in reactor20. When the expansion phase in reactor 10 is completed, the plasma inthat reactor returns to its precompression state (stage 0) while theplasma in reactor 20 completes the compression phase and is at stage 1.The plasma is reactor 10 is now maintained at stage 0 temporarily whilethe external magnetic field strength in reactor 20 is increasing so thatthe plasma volume in reactor 20 can be kept constant as the plasmatemperature is increasing. During this period, the power use to increasethe magnetic field strength in reactor 20 comes either from theelectrical power generated by the thermonuclear neutrons in reactor 10or from that in reactor 20 itself. When the plasma in reactor 20 reachesthermal balance (stage 2), it is allowed to expand. Some of theresulting work obtained from reactor 20 during its expansion phase istransferred via line 42 and used to compress the plasma in reactor 10.The balance of the energy is transferred to load 27 via line 36. Thecycle continues in this manner. The current in the coils 12 and 22 ofreactor 10 and 20 respectively as a function of time is illustrated inFIG. 8. Note that this coupled reactor system is analogous to a twocylinder internal combustion engine.

The averaged total electrical output power obtained from directconversion is defined as ##EQU39## Here P_(l) is the resistive powerdissipation in the coils, τ_(p) is the period of one complete cycle, andφdt is the cyclic time integral, i.e. time integration over the periodτ_(p).

Having generally described the invention, the following specific exampleis given as further illustration thereof. The power cycle with heatingphase at constant plasma volume is considered. Note that theneoclassical energy loss P_(neo), which could in principle be convertedinto electrically, is not included in Eq. (32). For a stelleratorreactor 10 with reactor parameters given in Table I, the thermalrelaxation time required by the plasma to reach stage 2 after theadiabatic compression with r_(v) =1/0.6 is approximately 0.16 sec. Ifthe compression and expansion phases are each carried out in 0.01 sec(note that τ_(i) =0(10⁻³) sec for the plasma with parameters given inTable I), then τ_(p) ≈0.18 sec. With this information and using Eq.(12), it can be shown that W_(net) /τ_(p) =2.2 GW.

If the plasma radius remains constant, then at r=a, the thermonuclearneutron power P_(n) =12.2 GW. The electrical power obtained from theneutrons at a conversion efficiency η_(c) =1/3 is η_(c) P_(n) =4.1 GW.Thus, W_(net) /τ_(p) is about 50% of η_(c) P_(n). Note that W_(net)/τ_(p) for each reactor in the coupled stellarator system may be lowerthan the 2.2 GW since τ_(p) for the coupled system is longer (see FIG.8). Also note that the actual values of W_(net) /τ_(p) and P_(n) shouldbe lower than the results obtained here since the radial profiles ofdensity and temperature must be taken into account properly.

To calculate the resistive power dissipation in the coils, 12, note that##EQU40## The factor T_(room) /T_(coil) is an approximation for thethermodynamic efficiency of the refrigeration cycle, i.e. ##EQU41##where T_(room) and T_(coil) are the room temperature and the cryogenictemperature of the cooled coils, respectively. The current in the coils12 is given by ##EQU42## where c is the speed of light and N is thenumber of turns of the coil per meter. The coil resistance is expressedas ##EQU43## where l = 2πa_(c) LN is the total length of the coil (a_(c)is the distance between the coil and the center of the plasma), A is thecoil cross sectional area, and η is the resistivity. If b is the widthof the coil (measured along the direction of the minor radius), thenA=b/N. Thus, Eq. (33) can be written as ##EQU44## Using the reactorparameters in Table I and setting a_(c) =3 m, b=0.5 m, T_(room) =293°K., T_(coil) =77° K. (the boiling point for liquid nitrogen), it can beshown that the total resistive power dissipation in a copper coil duringa complete cycle is on the order of a few percent of W_(net). Hence,φP_(l) dt can be neglected and P_(direct) =2.2 GW.

It may be thought that more neutron power could be obtained byincreasing the confining magnetic field strength 14 and operating theplasma 16 at a higher pressure without pulsing the minor radius.However, this mode of operation is undesirable since the neutron wallload would exceed the present day engineering limit which is below 10MW·m⁻².

A power cycle for direct conversion of alpha-particle energy intoelectricity for stellarator reactors has been described. The directenergy conversion is achieved by alternating compression and expansionof the plasma minor radius.

For the stellarator reactor with parameters given in Table I, theaveraged power obtained from the cycle with compression ratio 1/0.6 isabout 50% of the electrical power obtained from the thermonuclearneutrons from the same reactor without compressing the plasma. Thus, thecycle provides an alternative scheme for extracting energy from D-Tfueled reactors. This scheme can be used either alone or as a supplementto the electrical power generated from thermonuclear neutrons. Foradvanced neutron-lean fueled reactors, this power cycle may become animportant scheme for energy conversion.

The foregoing description of a preferred embodiment of the invention hasbeen presented for purposes of illustration and description. It is notintended to be exhaustive or to limit the invention to the precise formdisclosed, and obviously many modifications and variations are possiblein light of the above teaching. The embodiment was chosen and describedin order to best explain the principles of the invention and itspractical application to thereby enable others skilled in the art tobest utilize the invention in various embodiments and with variousmodifications as are suited to the particular use contemplated. It isintended that the scope of the invention be defined by the claimsappended hereto.

The embodiments of the invention in which an exclusive property orprivilege is claimed are defined as follows:
 1. In a stellarator fusionreactor having a fusion plasma in thermal balance disposed therein andexternal toroidal magnetic field coils and helical stabilizing coilssaid fusion plasma including alpha-particles and occupying a volume V₀and said magnetic field coils producing a toroidal confining magneticfield, a method of generating electricity in said external toroidalmagnetic fields coil from the energy of said alpha particles,(a)compressing said plasma adiabatically to a volume V₁ ; (b) maintainingthe volume of said compressed plasma at said volume V₁ for a timesubstantially equal to a thermal relaxation time, such that thetemperature of said plasma is driven up by thermonuclear alpha-particleheating thereby providing a heated plasma; and (c) expanding said heatedplasma to said volume V₀ and simultaneously generating, electricalenergy in said external toroidal field coils by a back-voltage producedby said expanding plasma.
 2. The method of claim 1 wherein the step ofcompressing said plasma is performed by increasing the toroidalconfining magnetic field produced by said toroidal magnetic field coils.3. The method of claim 2 wherein the step of maintaining the volume ofsaid compressed plasma is performed by further increasing the toroidalconfining magnetic field produced by said toroidal magnetic field coils.4. The method of claim 3 wherein the step of expanding said compressedplasma is performed by reducing the toroidal confining magnetic fieldproduced by said toroidal magnetic field coils.
 5. The method of claim 4wherein said method is repeated cyclically.
 6. In a stellarator fusionreactor having a fusion plasma in thermal balance disposed therein andexternal magnetic field coils and helical stabilizing coils said fusionplasma including alpha-particles and occupying a volume V₀ and saidexternal toroidal magnetic field coils producing as toroidal confiningmagnetic field, a method of generating electricity in said externaltoroidal magnetic field coils from the energy of said alpha particles,said method comprising the steps of:(a) compressing said plasmaadiabatically to a volume V₁ ; (b) expanding said compressed plasma, atconstant pressure, to a volume V₂ at which β=β_(c), said volume V₂ beinggreater than V₁ and less than V₀, such that temperature of said plasmais driven up by thermonuclear alpha-particle heating thereby providing aheated plasma; and (c) expanding said heated plasma from said volume V₂to said volume V₀, said β remaining at a value equal to β_(c) ; wherebyelectrical energy is simultaneously generated in said external toroidalmagnetic field coils by a back-voltage produced by said expandingplasma.
 7. The method of claim 6 wherein the step of compressing saidplasma is performed by increasing the toroidal confining magnetic fieldproduced by said toroidal magnetic field coils.
 8. The method of claim 7wherein the step of expanding said compressed plasma at constantpressure is performed by reducing the toroidal confining magnetic fieldproduced by said toroidal magnetic field coils at a rate sufficient tomaintain said plasma at a constant pressure.
 9. In a stellarator fusionreactor system having a first and a second stellarator reactor, each ofsaid first and second stellarator reactors having a fusion plasmadisposed therein and external toroidal magnetic field coils andstabilizing helical coils, said fusion plasma including alpha-particlesand occupying a volume V₀ and said toroidal magnetic field coilsproducing a toroidal confining magnetic field, a method of generatingelectricity in said external toroidal magnetic field coils from theenergy of said alpha-particles, said method comprising:(a) compressingthe plasma in said first reactor adiabatically to a volume V₁ ; (b)maintaining the volume of said compressed plasma in said first reactorat said volume V₁ for a time substantially equal to a thermal relaxationtime, such that the temperature of said plasma is driven up bythermonuclear alpha-particle heating thereby providing a heated plasma;(c) expanding said heated plasma in said first reactor to said volume V₀and simultaneously generating electrical energy in said externaltoroidal magnetic field coils of said first reactor by a back-voltageproduced by said expanding plasma; (d) transferring a portion of saidgenerated electrical energy to said second stellarator reactor in anamount sufficient to compress the plasma in said second reactor to avolume V₁ ; (e) compressing the plasma in said second reactoradiabatically to a volume V₁ ; (f) maintaining the volume of said plasmain said second reactor at said volume V₁, for a time substantially equalto a thermal relaxation time, such that the temperature of said plasmais driven up by thermonuclear alpha-particle heating thereby providing aheated plasma, while maintaining the volume of said plasma in said firstreactor at said volume V₀ ; and (g) expanding said heated plasma in saidsecond reactor to said volume V₀ and simultaneously generatingelectrical energy in said external toroidal magnetic field coils of saidsecond reactor by a back-voltage produced by said expanding plasma. 10.The method of claim 9 further including the steps of:transferring aportion of the electrical energy generated in said second reactorexternal toroidal magnetic field coils to said first reactor in anamount sufficient to compress the plasma in said first reaction to avolume V₁.
 11. The method of claim 10 wherein said method is repeatedcyclically.
 12. The method of claim 11 wherein steps (a) and (e) areperformed by increasing the toroidal confining magnetic field producedby said first and said second reactor toroidal magnetic field coils. 13.The method of claim 12 wherein steps (b) and (f) are performed byfurther increasing the toroidal confining magnetic field produced bysaid first and said second reactor toroidal magnetic field coils. 14.The method of claim 13 wherein steps (c) and (g) are performed byreducing the toroidal confining magnetic field produced by said firstand said second reactor toroidal magnetic field coils.
 15. A magneticconfinement fusion reactor for generating electricity comprising:(a)toroidal vacuum vessel having a fusion plasma disposed therein, saidplasma including alpha-particles; (b) helical magnetic stabilizing coilsdisposed about said vacuum vessel; (c) external toroidal magneticconfining field coils disposed about said vacuum vessel; (d) means forgenerating a current through said toroidal coils; (e) means forcompressing said plasma; (f) means for maintaining the volume of saidcompressed plasma constant, said means operable to maintain saidcompressed volume for a time substantially equal to a thermal relaxationtime; (g) means for expanding said plasma; and (h) means fortransmitting current from said toroidal coils generated by said plasma.16. The reactor of claim 15 wherein said means for compressing saidplasma comprises means for increasing the current through said toroidalcoils.
 17. The reactor of claim 16 wherein said means for maintainingthe volume of said compressed plasma comprises means for furtherincreasing the current through said toroidal coils.
 18. The reactor ofclaim 17 wherein the means for expanding said plasma comprising meansfor reducing the current through said toroidal coils.
 19. An electriccurrent generating system comprising:(a) a first toroidal vacuum vessel,said vacuum vessel having a fusion plasma disposed therein, said fusionplasma including alpha-particles; (b) A first set of helical magneticstabilizing coils disposed about said first toroidal vacuum vessel; (c)a first set of external toroidal magnetic coils disposed about saidfirst toroidal vacuum vessel, said first set of toroidal coils operableto generate a magnetic field, to confine and compress said plasma, whenan electric current is passed therethrough; (d) means for generating acurrent through said first set of toroidal coils; (e) means fortransmitting electrical current, generated by said plasma, from saidfirst set of toroidal coils; (f) a second toroidal vacuum vessel, saidvacuum vessel having a fusion plasma disposed therein, said fusionplasma including alpha-particles; (g) a second set of helical magneticstabilizing coils disposed about said second toroidal vacuum vessel; (h)a second set of external toroidal magnetic coils disposed about saidsecond toroidal vacuum vessel, said second set of toroidal coilsoperable to generate a magnetic field, to confine and compress saidplasma, when an electric current is passed therethrough; (i) means fortransmitting electric current, generated by said plasma, from saidsecond set of toroidal coils; (j) means for transferring electricalcurrent from said first set of toroidal coils to said second set oftoroidal coils. (k) means for transferring electric current from saidsecond set of toroidals coils to said first set of toroidal coils.